Helix to Straight Line

Given a straight line track and a helical track, we can combine them to improve the helix parameter measurements as we did for the helix to helix linking. Again the pivots should be chosen at a common space point: the best choice is the intersection of the straight line track with the boundary of the two regions. We then correct the error matrix of the straight line track for multiple scattering. Then the $\chi^2$ to minimize is defined by

$$\begin{array} {lll} \chi^2 = \Delta{\bf a}_H^T \cdot E_{{\... ...S^T \cdot E_{{\bf a}_S}^{-1} \cdot \Delta{\bf a}_S\end{array}$$ (24)

with

$$\begin{array} {lll} \nonumber \Delta {\bf a}_H & = & {\bf a... ... a} \cr \Delta {\bf a}_S & = & {\bf a}_S - {\bf a},\end{array}$$

where ${\bf a}_H$ and $E_{{\bf a}_H}$ are the helix parameter vector and its error matrix, while ${\bf a}_S$ and $E_{{\bf a}_S}$ are derived from those of the straight line (${\bf b}$ and $E_{\bf b}$) as follows:  

$$\begin{array} {lll} \begin{array} {lll} {\bf b} = (b_1,b_2,... ...\end{array} & * \end{array} \right).\end{array} \end{array}$$ (25)

Then what follows is the same as with the helix to helix linking in the previous subsection: it reduces to the two-track case: i = H,  S in Eq.2.5.31.